Stability of kernel bundles on projective bundles over curves

Abstract

Let $X$ be a projective bundle over a smooth curve $C$ of genus $g\ge 3$, and consider the relative hyperplane bundle ${\mathcal{O}}_X(1)\to X$. Let $V\subseteq H^0(X,{\mathcal{O}}_X(1))$ be a generating subspace. We prove that when $C$, $X$ and $V$ are general in moduli and ${\mathcal{O}}_X(1)$ is sufficiently ample, the kernel bundle of the system $({\mathcal{O}}_X(1),V)$ is ${\mathcal{O}}_X(1)$-stable.

10 pages